A multiple-volume book project: "Kuranishi spaces and Symplectic
Geometry"
I am writing a series of books titled "Kuranishi spaces and Symplectic
Geometry", which develops the foundations for those areas of Symplectic
Geometry which involve "counting" moduli spaces of J-holomorphic curves
--
Gromov-Witten invariants, Quantum Cohomology, Lagrangian Floer
cohomology, Fukaya categories, Symplectic Field Theory, and so on. I
hope that eventually there will be four or more volumes, which will go
at least as far as full definitions of Gromov-Witten theory, Quantum
Cohomology, Lagrangian Floer cohomology, and Fukaya categories.
Currently only volumes I and II are partially written, which develop
the theory of Kuranishi spaces, and you can download them from the
links below. I will continue to post updates on this page as I write
more. But the whole thing will take some years, so don't hold your
breath.
'Kuranishi
spaces and Symplectic Geometry. Volume I. Basic theory
of (m-)Kuranishi spaces',
pdf file.
'Kuranishi spaces
and Symplectic Geometry. Volume II. Differential
Geometry of (m-)Kuranishi spaces',
pdf file.
'Kuranishi spaces
and Symplectic Geometry. Volumes I and II together',
pdf file.
Note: if
you are reading the books on a computer, you may prefer the
combined file of volumes I and II, as there are clickable links made
using hyperref, including cross-references between volumes I and II.
All comments on the project are welcome.
Here is
a longer description of the project:
"Kuranishi spaces" are a class of geometric spaces introduced in
1990 by Fukaya and Ono, as the geometric structure on moduli spaces of
J-holomorphic curves in a symplectic manifold, and used in the work of
Fukaya, Oh, Ohta and Ono on Lagrangian Floer cohomology and Fukaya
categories. Although their definition was sufficient for their
applications, it did not give a very satisfactory notion of geometric
space -- notions of morphisms are not well behaved -- so the theory of
Kuranishi spaces was never developed very far.
Around 2006 I became interested in the question of what is
the "right" definition of Kuranishi space, which I felt
Fukaya-Oh-Ohta-Ono had not found. In 2008 I understood the first part
of the
answer, when I read David Spivak's thesis on 'derived
manifolds'. Scales fell from my eyes, and I immediately saw that:
Kuranishi
spaces are really derived smooth orbifolds, where 'derived' is in the
sense of the Derived Algebraic Geometry of Jacob Lurie and Toën-Vezzosi.
This has
important consequences: we should define and study Kuranishi spaces
using the ideas and methods of Derived Algebraic Geometry, including in
particular higher categories, and sheaves / higher sheaves. As
Fukaya-Oh-Ohta-Ono invented their Kuranishi spaces some years before
Derived Algebraic Geometry began, they lacked essential tools.
For my first version of a new definition of Kuranishi space, I
took what was available on derived diffential geometry in the work of
Lurie and Spivak and simplified it, to work in 2-categories rather than
∞-categories. The result was my theory of d-manifolds and d-orbifolds,
which you can read about on another page on
my
website. This gives strict 2-categories dMan and dOrb of 'd-manifolds' and
'd-orbifolds', "derived "versions of smooth manifolds and smooth
orbifolds, and also 2-categories dMan^{b}, dOrb^{b} of d-manifolds and d-orbifolds with
boundary, and 2-categories dMan^{c},
dOrb^{c} of d-manifolds and d-orbifolds with corners.
All these are defined as examples of derived
C^{∞}-schemes
and derived C^{∞}-stacks,
kinds of derived scheme and derived stacks over smooth functions. I
worked on this theory over 2008-2012. I am revising a book on it, to be
published by OUP, circa 2018.
In 2014 I realized that there is an alternative way to define
Kuranishi spaces, much closer to the original Fukaya-Oh-Ohta-Ono
definition, equivalent to my 2-categories to my 2-categories dMan and dOrb
of 'd-manifolds' and 'd-orbifolds'. To see why there should be two
definitions, consider the following two equivalent definitions of
manifolds:
- A
manifold is a Hausdorff, second countable topological space X equipped with an atlas of
pairwise compatible charts {(V_{i},ψ_{i}): i∈I}.
- A
manifold is a Hausdorff, second countable topological space X equipped with a sheaf O_{X} of ℝ-algebras
or C^{∞}-rings
locally isomorphic to (ℝ^{n},O_{ℝ}^{n}).
If we try to define derived manifolds by generalizing definition 1, we
get Kuranishi spaces (or something similar). If we try to define them
by generalizing definition 2, we get something like Spivak's derived
manifolds, or my d-manifolds.
I wrote up my new definition of Kuranishi spaces in arXiv:1409.6908, surveyed in
arXiv:1510.07444. These
define an ordinary category μKur
of 'μ-Kuranishi spaces' which
is equivalent to the homotopy category Ho(dMan) of the strict 2-category of d-manifolds dMan, and a weak 2-category mKur
of 'm-Kuranishi spaces' which is equivalent to dMan, and a
weak 2-category Kur of
'Kuranishi spaces', which is equivalent to the
strict 2-category dOrb of
d-orbifolds. They also defines boundary and corner versions μKur^{b}, μKur^{c},
mKur^{b}, mKur^{c}, Kur^{b}, Kur^{c}.^{
} In the book series, I take the point of view that
the definition of Kuranishi spaces takes as input a category Ḿan of 'manifolds' satisfying some
axioms, and gives as output (2-)categories μḰur,mḰur,Ḱur
of '(μ-and m-)Kuranishi spaces'. Here Ḿan
could be the usual category of classical manifolds Man, and then μḰur,mḰur,Ḱur
would be the usual (2-)categories μKur,mKur,Kur. But there are lots of other
possibilities for Ḿan,
including the category of manifolds with corners Man^{c }and
many variations of this, categories of manifolds with singularities,
and so on. So we define many (2-)categories of (μ-and m-)Kuranishi
spaces. In the C^{∞}-algebraic
geometry approach this would be much harder to do, as the category of
classical manifolds Man is
built into the definition of C^{∞}-ring, at the very bottom level of the
theory.
I am currently planning four or more volumes in the series, as
follows:
Volume I. Basic theory of
(m-)Kuranishi spaces. Definitions of the category μḰur of μ-Kuranishi spaces, and the
2-categories mḰur
of m-Kuranishi spaces and Ḱur of Kuranishi spaces, over a
category of 'manifolds' Ḿan
such as classical manifolds Man
or manifolds with corners Man^{c }.
Boundaries, corners, and corner (2-)functors for (μ-and m-)Kuranishi
spaces with corners. Relation to similar structures in the literature,
including Fukaya-Oh-Ohta-Ono's Kuranishi spaces, and
Hofer-Wysocki-Zehnder's polyfolds. 'Kuranishi moduli problems', our
approach to putting Kuranishi structures on moduli spaces, canonical up
to equivalence.
Volume II. Differential
Geometry of (m-)Kuranishi spaces. Tangent and obstruction spaces for
(μ-and m-)Kuranishi spaces. Canonical bundles and orientations.
(W-)tran sversality, (w-)submersions, and existence of w-transverse
fibre products in mḰur
and Ḱur. M-(co)homology of manifolds and
orbifolds as in my arXiv:1509.05672,
virtual (co)chains and virtual (co)cycles for compact, oriented
(m-)Kuranishi spaces in M-(co)homology. Orbifold strata of Kuranishi
spaces. Bordism and cobordism for (m-)Kuranishi spaces.
Volume III. Kuranishi
structures on moduli spaces of J-holomorphic
curves. For very many moduli spaces of J-holomorphic curves M
of interest in Symplectic Geometry, including singular curves, curves
with Lagrangian boundary conditions, marked points, etc., we show that M can be made into a Kuranishi
space M,
uniquely up to equivalence in Ḱur.
We do this by a new method using 2-categories, similar to
Grothendieck's 'representable functor' approach to moduli spaces in
Algebraic Geometry. We do the same for many other classes of moduli
problems for nonlinear elliptic p.d.e.s, including gauge theory moduli
spaces. Natural relations between moduli spaces, such as maps F_{i} : M_{k}_{+1} → M_{k} forgetting a marked
point, correspond to relations between the Kuranishi spaces, such as a
1-morphism F_{i} : M_{k}_{+1}
→ M_{k}
in Ḱur. We discuss orientations on
Kuranishi moduli spaces.
Volumes IV-. Big theories in
Symplectic Geometry. To include Gromov-Witten invariants, Quantum
Cohomology, Lagrangian Floer cohomology, and Fukaya categories.
References
K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, 'Lagrangian
intersection Floer theory - anomaly and
obstruction', Parts I and II. AMS/IP Studies in Advanced
Mathematics,
46.1 and 46.2, A.M.S./International Press, 2009.
K. Fukaya and K. Ono, 'Arnold Conjecture and
Gromov-Witten invariant', Topology 38 (1999), 933-1048.
D. Joyce, 'Algebraic Geometry over C^{∞}-rings', to appear in
Memoirs of the A.M.S.
-- Also available on the Web as arXiv:1001.0023.
D. Joyce, 'An introduction to C^{∞}-schemes
and
C^{∞}-algebraic
geometry', pages 299-325 in
H.-D. Cao and S.-T. Yau, editors, Surveys in Differential Geometry 17
(2012), Lectures given at the JDG symposium, June 2010, in memory of
C.C. Hsiung.
-- Also available on the Web as
arXiv:1104.4951.
D. Joyce, 'An introduction to d-manifolds and derived
differential geometry', pages 230-281 in L. Brambila-Paz, O.
Garcia-Prada, P. Newstead and R.P. Thomas, editors, 'Moduli spaces', London Mathematical
Society Lecture Note Series 411, Cambridge University Press, 2014.
-- Also available on the Web as arXiv:1206.4207.
D. Joyce, 'D-manifolds, d-orbifolds and derived differential
geometry: a detailed summary', arXiv:1208.4948, 2012.
D. Joyce, 'D-manifolds and d-orbifolds: a theory of derived
differential geometry', preliminary version of book (2012)
pdf file. 768
pages.
D. Joyce, 'A new definition of Kuranishi space', arXiv:1409.6908, 2014.
D. Joyce, 'Some new homology and cohomology theories of manifolds
and orbifolds', arXiv:1509.05672,
2015.
D. Joyce, 'Kuranishi spaces as a 2-category', arXiv:1510.07444, 2015.
D.I. Spivak, 'Derived smooth manifolds',
Duke Mathematical Journal 153 (2010), 55-128. arXiv:0810.5174.